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One of U1 and U2 is a subspace of V while the other is not. Determine which is which and explain the criteria for subspace inclusion.

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Final answer:

To determine if U1 or U2 is a subspace of V, we need to check for closure under addition, scalar multiplication, and containing the zero vector.

Step-by-step explanation:

In order to determine whether U1 or U2 is a subspace of V, we need to check if they satisfy three criteria: closure under addition, closure under scalar multiplication, and containing the zero vector.

To check closure under addition, we need to verify if for any two vectors in U1/U2, their sum is also in U1/U2.

To check closure under scalar multiplication, we need to verify if for any scalar c and any vector in U1/U2, the product of c and the vector is also in U1/U2.

Finally, to check if U1/U2 contains the zero vector, we need to ensure that the zero vector is in U1/U2.

If U1 satisfies all three criteria, then U1 is a subspace of V. Otherwise, U2 is the subspace of V.

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